Implication Details

Claim: Given a morphism whose category has pushouts, if it is an extremal monomorphism, then it is a strong monomorphism.

Proof: Let m:A→Bm : A \to B be an extremal monomorphism and consider a diagram C→eDf↓↓gA→mB\begin{CD} C @>{e}>> D \\ @V{f}VV @VV{g}V \\ A @>>{m}> B \end{CD} in which e:C→De : C \to D is an epimorphism. Choose a pushout C→eDf↓↓uA→vP.\begin{CD} C @>{e}>> D \\ @V{f}VV @VV{u}V \\ A @>>{v}> P. \end{CD} Here, vv is an epimorphism since ee is an epimorphism. Moreover, by the universal property of the pushout, there is a unique morphism h:P→Bh : P \to B such that h∘v=mh \circ v = m and h∘u=gh \circ u = g. Since mm is an extremal monomorphism, vv is an isomorphism. Then v−1∘u:D→Av^{-1} \circ u : D \to A is the required filling of the first diagram, since v−1∘u∘e=v−1∘v∘f=f.v^{-1} \circ u \circ e = v^{-1} \circ v \circ f = f.